THE BANACH - MAZUR DISTANCE BETWEEN THE TRACE CLASSES c /

The Banach-Mazur distance between l2 ® l2 and l2 ® l2 is shown to be of the order Vmin(«,m) . Our proof yields that the distance between the trace classes c¡¡ and c¡¡ is of the same order as d(l£, l£). In this note we determine the distances between some tensor products of Euclidean spaces l2, k = 1, 2 ... . Let E, F be finite dimensional Banach spaces over the real field. The Banach-Mazur distance d(E, F) is defined as inf {|| T\\ \\T~X\\ | T is an isomorphism from E onto F j. In this note by E <Ê> F [resp. E ® F] we denote the algebraic tensor product E ® F endowed with the greatest [resp. the least] norm such that ||e ® /|| = ||e|| y/11 for e E E,f E F. The space /2" ® /2m with the norm / \'/2 s«,//®/; = Ski2 . ij V ij I where {ex, . . . , e„), {/,, . . . ,/m) are orthonormal bases for l2 and l2 respectively, is denoted by HS(/2, l2) or simply HS. Theorem 1. Let n, m be positive integers with n < m. Then (2\Te )~lVn~ < d(l!¿ ® I?, /2" ê /2m ) < 10\6ï . Proof. We begin with the upper estimate of the distance d(l2 ® l2, l2 ® l2). The argument given below works only for n > 36. However if n < 36 and i: l2 <è l2 -* l2 ® l2 denotes the formal identity map, then ||/|| < 1 and ||i_1|| < n < 10V« . We shall construct the isomorphism T: /2" <8> l2 -+ l2 ® l2 in the form T = j*°u°j where/: /2" ® l? -* HS(/2", /2m) is the natural embedding and u is an isometry of the «w-dimensional Hilbert space HS(/2, l2). It is easy to check that ||/_1|| < Vñ . Since (/*)"' = (/"')♦ we obtain H7'_1ll<IU"Il|-||"~1lMO,T,l<»Thus the proof of the upper estimate will be complete if we find a T with ||r|| < 10/V« . This is done in the following proposition. Received by the editors January 3, 1978. AMS (MOS) subject classifications (1970). Primary 46B99.