Involutive triangular matrix algebras

Abstract

In this paper we provide new examples of Banach $ \ast $-subalgebras of the matrix algebra $M_n(\mathscr{A}) $ over a commutative unital $C^*$-algebra $\mathscr{A}$. For any involutive algebra, we define two involutions on the triangular matrix extensions. We prove that the triangular matrix algebras over any commutative unital $C^*$-algebra are Banach ${\ast}$-algebras and that the primitive ideals of these algebras and some of their Banach $ \ast $-subalgebras are all maximal.

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Keywords

• primitive ideal
• maximal ideal
• Banach ∗-algebra
• C*-algebra

References

1. [1] O.M. Di Vincenzo, P. Koshlukov and R. La Scala, Involutions for upper triangular matrix algebras, Adv. in Appl. Math. 37, 541–568, 2006.
2. [2] J. Dixmier, C*-Algebras, North-Holland, Amsterdam, 1977.
3. [3] N. Jacobson, A topology for the set of primitive ideals in an arbitrary ring, Proc, Nat, Acad, Sci. U.S.A. 31, 333–338, 1945.
4. [4] T.K. Lee and Y. Zhou, Armendariz and reduced rings, Comm. Algebra 32 (6), 2287– 2299, 2004.
5. [5] G.J. Murphy, C*-Algebras and Operator Theory Academic Press, 1990.
6. [6] T.W. Palmer, Banach Algebras and the General Theory of $\ast$-Algebras Volume I Al- gebras and Banach Algebras, Encyclopedia of Mathematics and its Applications, Vol. 1, 1994.
7. [7] V. Paulsen, Completely Bounded Maps and Operator Algebras, vol. 78, Cambridge University Press, 2002.