I've installed banshee using these commands:

sudo add-apt-repository ppa:banshee-team/ppa
sudo apt-get update
sudo apt-get install banshee
And these are my results:

user@user-laptop:~$ sudo add-apt-repository ppa:banshee-team/banshee-daily
[sudo] password for user:
Executing: gpg --ignore-time-conflict --no-options --no-default-keyring --secret-keyring /etc/apt/secring.gpg --trustdb-name /etc/apt/trustdb.gpg --keyring /etc/apt/trusted.gpg --primary-keyring /etc/apt/trusted.gpg --keyserver keyserver.ubuntu.com --recv 9D2C2E0A3C88DD807EC787D74874D3686E80C6B7
gpg: requesting key 6E80C6B7 from hkp server keyserver.ubuntu.com
gpg: key 6E80C6B7: public key "Launchpad PPA for Banshee Team" imported
gpg: no ultimately trusted keys found
gpg: Total number processed: 1
gpg: imported: 1 (RSA: 1)
user@user-laptop:~$ sudo apt-get update
Ign cdrom://Ubuntu 11.04 _Natty Narwhal_ - Release i386 (20110427.1)/ natty/main Translation-en_US
Ign cdrom://Ubuntu 11.04 _Natty Narwhal_ - Release i386 (20110427.1)/ natty/restricted Translation-en_US
Hit http://us.archive.ubuntu.com lucid Release.gpg
Ign http://us.archive.ubuntu.com/ubuntu/ lucid/main Translation-en_US
Ign http://us.archive.ubuntu.com/ubuntu/ lucid/restricted Translation-en_US
Ign http://us.archive.ubuntu.com/ubuntu/ lucid/universe Translation-en_US
Ign http://us.archive.ubuntu.com/ubuntu/ lucid/multiverse Translation-en_US
Get:1 http://us.archive.ubuntu.com lucid-updates Release.gpg [198B]
Ign http://us.archive.ubuntu.com/ubuntu/ lucid-updates/main Translation-en_US
Ign http://us.archive.ubuntu.com/ubuntu/ lucid-updates/restricted Translation-en_US
Ign http://us.archive.ubuntu.com/ubuntu/ lucid-updates/universe Translation-en_US
Ign http://us.archive.ubuntu.com/ubuntu/ lucid-updates/multiverse Translation-en_US
Hit http://us.archive.ubuntu.com lucid Release
Get:2 http://us.archive.ubuntu.com lucid-updates Release [58.3kB]
Hit http://archive.canonical.com lucid Release.gpg
Ign http://archive.canonical.com/ lucid/partner Translation-en_US
Get:3 http://ppa.launchpad.net lucid Release.gpg [316B]
Ign http://ppa.launchpad.net/banshee-tea...-daily/ubuntu/ lucid/main Translation-en_US
Get:4 http://security.ubuntu.com lucid-security Release.gpg [198B]
Ign http://security.ubuntu.com/ubuntu/ lucid-security/main Translation-en_US
Ign http://security.ubuntu.com/ubuntu/ lucid-security/restricted Translation-en_US
Ign http://security.ubuntu.com/ubuntu/ lucid-security/universe Translation-en_US
Ign http://security.ubuntu.com/ubuntu/ lucid-security/multiverse Translation-en_US
Hit http://archive.canonical.com lucid Release
Get:5 http://ppa.launchpad.net lucid Release [14.0kB]
Get:6 http://security.ubuntu.com lucid-security Release [57.3kB]
Hit http://us.archive.ubuntu.com lucid/main Packages
Hit http://us.archive.ubuntu.com lucid/restricted Packages
Hit http://us.archive.ubuntu.com lucid/main Sources
Hit http://us.archive.ubuntu.com lucid/restricted Sources
Hit http://us.archive.ubuntu.com lucid/universe Packages
Hit http://us.archive.ubuntu.com lucid/universe Sources
Hit http://us.archive.ubuntu.com lucid/multiverse Packages
Hit http://us.archive.ubuntu.com lucid/multiverse Sources
Get:7 http://us.archive.ubuntu.com lucid-updates/main Packages [658kB]
Hit http://archive.canonical.com lucid/partner Packages
Get:8 http://ppa.launchpad.net lucid/main Packages [13.7kB]
Get:9 http://security.ubuntu.com lucid-security/main Packages [461kB]
Get:10 http://us.archive.ubuntu.com lucid-updates/restricted Packages [4,630B]
Get:11 http://us.archive.ubuntu.com lucid-updates/main Sources [231kB]
Get:12 http://us.archive.ubuntu.com lucid-updates/restricted Sources [2,196B]
Get:13 http://us.archive.ubuntu.com lucid-updates/universe Packages [282kB]
Get:14 http://security.ubuntu.com lucid-security/restricted Packages [2,867B]
Get:15 http://security.ubuntu.com lucid-security/main Sources [132kB]
Get:16 http://us.archive.ubuntu.com lucid-updates/universe Sources [104kB]
Get:17 http://us.archive.ubuntu.com lucid-updates/multiverse Packages [11.5kB]
Get:18 http://us.archive.ubuntu.com lucid-updates/multiverse Sources [5,827B]
Get:19 http://security.ubuntu.com lucid-security/restricted Sources [1,267B]
Get:20 http://security.ubuntu.com lucid-security/universe Packages [138kB]
Get:21 http://security.ubuntu.com lucid-security/universe Sources [42.5kB]
Get:22 http://security.ubuntu.com lucid-security/multiverse Packages [5,353B]
Get:23 http://security.ubuntu.com lucid-security/multiverse Sources [2,350B]
Fetched 2,228kB in 20s (108kB/s)
Reading package lists... Done
W: Duplicate sources.list entry http://archive.canonical.com/ lucid/partner Packages (/var/lib/apt/lists/archive.canonical.com_dists_lucid_partner_binary-i386_Packages)
W: Duplicate sources.list entry http://archive.canonical.com/ lucid/partner Packages (/var/lib/apt/lists/archive.canonical.com_dists_lucid_partner_binary-i386_Packages)
W: You may want to run apt-get update to correct these problems
user@user-laptop:~$ sudo apt-get install banshee
Reading package lists... Done
Building dependency tree
Reading state information... Done
The following packages were automatically installed and are no longer required:
libgdata1.4-cil podsleuth libboo2.0.9-cil sdparm firefox-branding
libwebkit1.1-cil
Use 'apt-get autoremove' to remove them.
The following extra packages will be installed:
banshee-extension-ubuntuonemusicstore libgdata1.7-cil libgkeyfile1.0-cil
libgpod4 libgudev1.0-cil libubuntuone1.0-cil
Suggested packages:
banshee-dbg
The following NEW packages will be installed:
banshee-extension-ubuntuonemusicstore libgdata1.7-cil libgkeyfile1.0-cil
libgudev1.0-cil libubuntuone1.0-cil
The following packages will be upgraded:
banshee libgpod4
2 upgraded, 5 newly installed, 0 to remove and 123 not upgraded.
Need to get 4,494kB of archives.
After this operation, 1,606kB of additional disk space will be used.
Do you want to continue [Y/n]? y
Get:1 http://us.archive.ubuntu.com/ubuntu/ lucid/main libubuntuone1.0-cil 0.3.1-0ubuntu1 [7,466B]
Get:2 http://ppa.launchpad.net/banshee-tea...-daily/ubuntu/ lucid/main libgdata1.7-cil 1.7.0.1-1~lucid1 [298kB]
Get:3 http://ppa.launchpad.net/banshee-tea...-daily/ubuntu/ lucid/main libgkeyfile1.0-cil 0.1-1ubuntu1~hyper1+lucid [11.1kB]
Get:4 http://ppa.launchpad.net/banshee-tea...-daily/ubuntu/ lucid/main libgpod4 0.8.0-3~hyper1+lucid [263kB]
Get:5 http://ppa.launchpad.net/banshee-tea...-daily/ubuntu/ lucid/main libgudev1.0-cil 0.1-2~hyper1+lucid [8,346B]
Get:6 http://ppa.launchpad.net/banshee-tea...-daily/ubuntu/ lucid/main banshee 1.9.4+git20110228.r1.704ddb6-0ubuntu1+lucid [3,812kB]
Get:7 http://ppa.launchpad.net/banshee-tea...-daily/ubuntu/ lucid/main banshee-extension-ubuntuonemusicstore 1.9.4+git20110228.r1.704ddb6-0ubuntu1+lucid [93.8kB]
Fetched 4,494kB in 32s (139kB/s)
Selecting previously deselected package libgdata1.7-cil.
(Reading database ... 148644 files and directories currently installed.)
Unpacking libgdata1.7-cil (from .../libgdata1.7-cil_1.7.0.1-1~lucid1_all.deb) ...
Selecting previously deselected package libgkeyfile1.0-cil.
Unpacking libgkeyfile1.0-cil (from .../libgkeyfile1.0-cil_0.1-1ubuntu1~hyper1+lucid_all.deb) ...
Preparing to replace libgpod4 0.7.93-0ubuntu1 (using .../libgpod4_0.8.0-3~hyper1+lucid_i386.deb) ...
Unpacking replacement libgpod4 ...
Selecting previously deselected package libgudev1.0-cil.
Unpacking libgudev1.0-cil (from .../libgudev1.0-cil_0.1-2~hyper1+lucid_all.deb) ...
Preparing to replace banshee 1.6.1-1~lucid1 (using .../banshee_1.9.4+git20110228.r1.704ddb6-0ubuntu1+lucid_i386.deb) ...
Unpacking replacement banshee ...
Selecting previously deselected package libubuntuone1.0-cil.
Unpacking libubuntuone1.0-cil (from .../libubuntuone1.0-cil_0.3.1-0ubuntu1_all.deb) ...
Selecting previously deselected package banshee-extension-ubuntuonemusicstore.
Unpacking banshee-extension-ubuntuonemusicstore (from .../banshee-extension-ubuntuonemusicstore_1.9.4+git20110228.r1.704ddb6-0ubuntu1+lucid_i386.deb) ...
Processing triggers for hicolor-icon-theme ...
Processing triggers for man-db ...
Processing triggers for shared-mime-info ...
Processing triggers for python-gmenu ...
Rebuilding /usr/share/applications/desktop.en_US.utf8.cache...
Processing triggers for desktop-file-utils ...
Processing triggers for python-support ...
Setting up libgdata1.7-cil (1.7.0.1-1~lucid1) ...
* Installing 14 assemblies from libgdata1.7-cil into Mono

Setting up libgkeyfile1.0-cil (0.1-1ubuntu1~hyper1+lucid) ...
* Installing 1 assembly from libgkeyfile1.0-cil into Mono

Setting up libgpod4 (0.8.0-3~hyper1+lucid) ...

Setting up libgudev1.0-cil (0.1-2~hyper1+lucid) ...
* Installing 1 assembly from libgudev1.0-cil into Mono

Setting up banshee (1.9.4+git20110228.r1.704ddb6-0ubuntu1+lucid) ...

Setting up libubuntuone1.0-cil (0.3.1-0ubuntu1) ...
* Installing 1 assembly from libubuntuone1.0-cil into Mono

Setting up banshee-extension-ubuntuonemusicstore (1.9.4+git20110228.r1.704ddb6-0ubuntu1+lucid) ...
Processing triggers for libc-bin ...
ldconfig deferred processing now taking place
Apparently,the banshee file is a tar.gz type file and when I try to extract it in order to open it, it takes me to a menus with a list and I click on partial. When I click on the partial option and then I click on the extract button, this window will message appear:

Extraction not performed

You don't have the right permissions to extract archives in the folder "file:///var/cache/apt/archives/partial"
What am I doing wrong?