The gap between local multiplier algebras

MARTIN ARGERAMI; DOUGLAS FARENICK; PEDRO MASSEY

Lincoln-Estados Unidos

Congreso; Great plains in operator theory symposium; 2007

University of Nebraska

The local multiplier algebra $M_{ m loc}(A)$ of a C$^*$-algebra $A$has the property that $M_{ m loc }(A)subseteq M_{ m loc}(M_{ mloc}(A))$. In this talk we show that if A is the C*-algebra tensor product of C([0,1]) and K(H) - compact operators in a separable Hilbert space - then the inclusion above is proper. Since the second  local multiplier algebra is, in this case, the injective envelope of A we conclude that the first and second local multiplier algebras algebras are not isomorphic. We show this by computing explicitly a *-representation for the inclusion Ainc I(A), where I(A) denotes the injective envelope of A.